1.what if they changed the scoring system in American Football, and there were only two ways o score, a field goal worth 3 points and a touchdown worth 7 points. What are all the possible total scores that a team could have?
all multiples of 3 and 7, added together:
3,6,9,12,...
7,14,21,28,...
and sums of those rows in all combinations.
To find the possible total scores that a team could have under the given scoring system in American Football, where a field goal is worth 3 points and a touchdown is worth 7 points, we can analyze the problem using mathematics.
We can think of this as a combinatorial problem, where we need to find all the possible combinations of field goals and touchdowns that would result in a specific total score. Let's form a table to help us visualize this.
Let the number of touchdowns be represented by 'T' and the number of field goals be represented by 'F.' Given that a touchdown is worth 7 points and a field goal is worth 3 points, the total score can be calculated using the formula: Total Score = (7 * T) + (3 * F).
Now, let's evaluate the possible values for 'T' and 'F' to find all the different combinations. Since the total score could be any non-negative number, we can express any possible score as a combination of touchdowns and field goals.
Here, we'll assume 'n' touchdowns and 'm' field goals, where 'n' and 'm' lie between 0 and infinity, inclusive. Since we can have zero touchdowns or zero field goals, the possibilities are endless.
For a more specific answer, we can consider a reasonable range of touchdowns and field goals that would represent a typical range of scores in American Football. For example, let's consider a range of 0 to 10 touchdowns and 0 to 5 field goals.
Using this range, we can analyze the combinations and calculate the corresponding total scores:
- Possible combinations of touchdowns (T) and field goals (F):
* 0 touchdowns and 0 field goals: (0, 0) => Total Score = (7 * 0) + (3 * 0) = 0
* 0 touchdowns and 1 field goal: (0, 1) => Total Score = (7 * 0) + (3 * 1) = 3
* 0 touchdowns and 2 field goals: (0, 2) => Total Score = (7 * 0) + (3 * 2) = 6
...
* 10 touchdowns and 0 field goals: (10, 0) => Total Score = (7 * 10) + (3 * 0) = 70
* 10 touchdowns and 1 field goal: (10, 1) => Total Score = (7 * 10) + (3 * 1) = 73
* 10 touchdowns and 2 field goals: (10, 2) => Total Score = (7 * 10) + (3 * 2) = 76
...
* 10 touchdowns and 5 field goals: (10, 5) => Total Score = (7 * 10) + (3 * 5) = 105
Therefore, under the given scoring system, the possible total scores that a team could have range from 0 to 105, with 3-point intervals.
Please note that this analysis is based on the assumption of a reasonable range of touchdowns and field goals. In reality, there is no limit to the number of touchdowns or field goals, so the possible total scores are theoretically infinite.